This is not a test prep unit (although it could be construed as such). My goal is not to simply expose students to test questions and format (although they will get an amazingly wide variety of exposure and practice). I want to help my students look back on the year and fill in the gaps and build on their foundation through repeated practice. All students need a chance to recall the things they have forgotten (or never mastered in the first place). Our curricula often push forward, build up and look ahead. But we can’t forget to spiral back and give students an opportunity to revisit topics from past units. These lessons represent my attempt designing a rewarding and fulfilling review unit for algebra. It is an attempt to prepare students for the summative exams that we use to asses our curriculum design and mark student progress.
The lessons in this unit are built around a simple format:
The work reserved for the final 45 minutes of class is connected to a series of homework assignments. The materials are designed so that whatever students don’t finish in class, they can do at home. If they finish the current assignment, they can move ahead and complete future assignments.
Each day, the lesson starts with some follow up on their practice work. Based on the previous lesson, I will be prepared to ask specific local and global questions about their work. Local questions can be anything specific to a certain aspect of a problem, like "how did you get that number there?" Global questions can be things like, "how does this example connect to the problem before it?"
During our most recent school dance, we had a small dilemma: what should the DJ play? Every year students are disappointed. They "hate" the music and demand that we improve the playlist. So this year we used sampling methods to resolve the issue. We collected and analyzed the data. After the dance we had zero complaints. How is this possible?
This is how I start the discussion with the students. I want them to identify how many students did we need to survey, who those students were and where did we find them. Many students are confused at the start of this lesson, they are wondering why they didn't see this survey. In fact, the most common misunderstanding is about why we didn't just ask everyone.
I remind them that with a sufficient sample, there is no need to ask everyone.
"Well how did you make sure that everyone would be represented?"
I take ideas from the group and we agree that students from each grade and gender need to be represented. We also need to avoid asking biased groups, like "everyone on the football team." This means that a certain element of the survey selection needs to be random.
"What are some ways to choose random groups?"
"Classes are sorted somewhat randomly. So you could pick some classes from each grade level."
"That is one aspect of what we did. We asked 1 to 2 classes from each grade level (out of 4). What else do you think we did?"
Here we discuss the importance of randomness and the degrees of randomness of an event. Students agree that we might not be able to make something 100% random, but that we can certainly try things that are more random than not, like stopping students in the hallway at different times.
"The questions we ask are also important. We asked students for specific songs that they want to hear and other songs that they don't want to hear. Then we compared these choices for all of the responses and omitted songs that only appeared under the 'I don't want to hear this' column."
I then explain that this is qualitative univariate data, since students were simply listing what type of songs they did and did not want to hear. They can think of each song as its own category and then we count the frequency of mentions for wanting or not wanting to hear. I describe that we only used songs that students liked and were careful to arrange them in a way that "built up" momentum on the dance floor.
Perhaps the most surprising thing for students was that the sampling process actually worked for the dance. They agree that the music was the best it had ever been and are interested to learn that a mathematical approach was behind the success.
Then we turn to the practice problems at hand. Although they are much less fun, they revolve around the same concepts:
After the discussion around dance playlist, students are quick to point out that the bias here is in the concentration of younger participants. I ask them "how would you fix the survey question so that the data was not biased?" Students agree that if we omit data from certain age groups, we could ask something like "what are the driving habits of people under 35?"
We give students a set of data problems and a helpful template to follow along. The templates are set up to help me recognize when a student needs help. Here is a link to the Types of Data Template. In the template, my students rate how they feel about each problem as they finish and I look at these numbers to figure out if they feel comfortable with the material. I try and keep this rating system simple. Something like, give yourself a 4 if you really understood the question and give yourself a 1 if you felt really overwhelmed.